A Primer on Hilbert Space Theory: Linear Spaces, Topological Spaces, Metric Spaces, Normed Spaces, and Topological Groups (UNITEXT for Physics)

This book is an introduction to the theory of Hilbert space, a fundamental tool for non-relativistic quantum mechanics. Linear, topological, metric, and normed spaces are all addressed in detail, in a rigorous but reader-friendly fashion. The rationale for an introduction to the theory of Hilbert space, rather than a detailed study of Hilbert space theory itself, resides in the very high mathematical difficulty of even the simplest physical case. Within an ordinary graduate course in physics there is insufficient time to cover the theory of Hilbert spaces and operators, as well as distribution theory, with sufficient mathematical rigor. Compromises must be found between full rigor and practical use of the instruments. The book is based on the author's lessons on functional analysis for graduate students in physics. It will equip the reader to approach Hilbert space and, subsequently, rigged Hilbert space, with a more practical attitude. With respect to the original lectures, the mathematical flavor in all subjects has been enriched. Moreover, a brief introduction to topological groups has been added in addition to exercises and solved problems throughout the text. With these improvements, the book can be used in upper undergraduate and lower graduate courses, both in Physics and in Mathematics. Preface; Contents; Symbols; 1 Introduction and Preliminaries; 1.1 Hilbert Space Theory--A Quick Overview; 1.1.1 The Real Numbers--Where it All Begins; 1.1.2 Linear Spaces; 1.1.3 Topological Spaces; 1.1.4 Metric Spaces; 1.1.5 Normed Spaces and Banach Spaces; 1.1.6 Topological Groups; 1.2 Preliminaries; 1.2.1 Sets; 1.2.2 Common Sets; 1.2.3 Relations Between Sets; 1.2.4 Families of Sets; Union and Intersection; 1.2.5 Set Difference, Complementation, and De Morgan's Laws; 1.2.6 Finite Cartesian Products; 1.2.7 Functions; 1.2.8 Arbitrary Cartesian Products; 1.2.9 Direct and Inverse Images 1.2.10 Indicator Functions1.2.11 Cardinality; 1.2.12 The Cantor-Shröder-Bernstein Theorem; 1.2.13 Countable Arithmetic; 1.2.14 Relations; 1.2.15 Equivalence Relations; 1.2.16 Ordered Sets; 1.2.17 Zorn's Lemma; 1.2.18 A Typical Application of Zorn's Lemma; 1.2.19 The Real Numbers; References; 2 Linear Spaces; 2.1 Linear Spaces--Elementary Properties and Examples; 2.1.1 Elementary Properties of Linear Spaces; 2.1.2 Examples of Linear Spaces; 2.2 The Dimension of a Linear Space; 2.2.1 Linear Independence, Spanning Sets, and Bases; 2.2.2 Existence of Bases; 2.2.3 Existence of Dimension 2.3 Linear Operators2.3.1 Examples of Linear Operators; 2.3.2 Algebra of Operators; 2.3.3 Isomorphism; 2.4 Subspaces, Products, and Quotients; 2.4.1 Subspaces; 2.4.2 Kernels and Images; 2.4.3 Products and Quotients; 2.4.4 Complementary Subspaces; 2.5 Inner Product Spaces and Normed Spaces; 2.5.1 Inner Product Spaces; 2.5.2 The Cauchy-Schwarz Inequality; 2.5.3 Normed Spaces; 2.5.4 The Family of ellp Spaces; 2.5.5 The Family of Pre-Lp Spaces; References; 3 Topological Spaces; 3.1 Topology--Definition and Elementary Results; 3.1.1 Definition and Motivation; 3.1.2 More Examples 4.2 Topology and Convergence in a Metric Space4.2.1 The Induced Topology; 4.2.2 Convergence in Metric Spaces; 4.3 Non-Expanding Functions and Uniform Continuity; 4.4 Complete Metric Spaces; 4.4.1 Complete Metric Spaces; 4.4.2 Banach's Fixed-Point Theorem; 4.4.3 Baire's Theorem; 4.4.4 Completion of a Metric Space; 4.5 Compactness and Boundedness; References; 5 Normed Spaces; 5.1 Semi-Norms, Norms, and Banach Spaces; 5.1.1 Semi-Norms and Norms; 5.1.2 Banach Spaces; 5.1.3 Bounded Operators; 5.1.4 The Open Mapping Theorem; 5.1.5 Banach Spaces of Linear and Bounded Operators 3.1.3 Elementary Observations3.1.4 Closed Sets; 3.1.5 Bases and Subbases; 3.2 Subspaces, Point-Set Relationships, and Countability Axioms; 3.2.1 Subspaces and Point-Set Relationships; 3.2.2 Sequences and Convergence; 3.2.3 Second Countable and First Countable Spaces; 3.3 Constructing Topologies; 3.3.1 Generating Topologies; 3.3.2 Coproducts, Products, and Quotients; 3.4 Separation and Connectedness; 3.4.1 The Hausdorff Separation Property; 3.4.2 Path-Connected and Connected Spaces; 3.5 Compactness; References; 4 Metric Spaces; 4.1 Metric Spaces--Definition and Examples This introduction to Hilbert space will equip the reader for more demanding further study. It builds a pedagogic bridge between the rigorous style of mathematics and the more practical perspective of physicists